Research Article Open Access

On Hyperplanes of the Geometry D4,2 and their Related Codes

Abdelsalam Abou Zayda

Abstract

Problem statement: The point-line geometry of type D4,2 was introduced and characterized by many authors such as Shult and Buekenhout and in several researches many of geometries were considered to construct good families of codes and this forced us to present very important substructures in such geometry that are hyperplanes. Approach: We used the isomorphic classical polar space Ω+(8, F) and their combinatorics to construct the hyperplanes and the family of certain codes related to such hyperplanes. Results: We proved that each hyperplane is either the set Δ2 (p) which consisted of all points at a distance mostly 2 from a fixed point p or a Grassmann geometry of type A3,2 and then we presented a new family of non linear binary constant-weight codes. Conclusion: The hyperplanes of the geometry D4,2 allow us to discuss further substructures of the geometry such as veldkamp spaces.

Journal of Mathematics and Statistics
Volume 5 No. 1, 2009, 72-76

DOI: https://doi.org/10.3844/jmssp.2009.72.76

Submitted On: 7 May 2008 Published On: 31 March 2009

How to Cite: Zayda, A. A. (2009). On Hyperplanes of the Geometry D4,2 and their Related Codes. Journal of Mathematics and Statistics, 5(1), 72-76. https://doi.org/10.3844/jmssp.2009.72.76

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Keywords

  • Hyperplane
  • grassmann geometry
  • constant weight code